Pentatrigesimal

Regarding the previous list which I recently posted, the answer is that the list was prime numbers up to 42,875 listed in base 35. 42,875 is 35 cubed and in base 35 is of course 1000.

The logic behind base 35, is, as you might gather by looking at the table, to write numbers using all numbers and all letters of the English alphabet except O. The reason for not using O is, of course, that it is so similar to zero and that casually zero is often read as “O”. In fact, when I was a child, I was often told that “O is a letter, and zero [0] is the number” whenever I read the number as “O”, but would reply that “O is a number”.

The properties of this system of all numbers and all letters except O, scientifically called “pentatrigesimal” are both familiar and unfamiliar compared to decimal.

Like ten, 35 is a deficient semiprime (=5x7). However, 35 is a much more deficient number than ten, having a deficiency of 22 (or 63 percent) as against 2 for the number 10. Thus, the proportion of fractions that terminate in pentatrigesimal is much smaller than in decimal: only denominators whose only prime factors are 5 and 7 terminate, and there are only 38 such numbers smaller than 1,500,625 (35⁴, or 10000 in pentatrigesimal):

Decimal	Pentatrigesimal	Prime factorisation
5	5	5
7	7	7
25	Q	5*5
35	10	5*7
49	1E	7*7
125	3K	5*5*5
175	50	5*5*7
245	70	5*7*7
343	9T	7*7*7
625	HV	5*5*5*5
875	Q0	5*5*5*7
1,225	100	5*5*7*7
1,715	1E0	5*7*7*7
2,401	1YL	7*7*7*7
3,125	2JA	5*5*5*5*5
4,375	3K0	5*5*5*5*7
6,125	500	5*5*5*7*7
8,575	700	5*5*7*7*7
12,005	9T0	5*7*7*7*7
15,625	CRF	5*5*5*5*5*5
16,807	DQ7	7*7*7*7*7
21,875	HV0	5*5*5*5*5*7
30,625	Q00	5*5*5*5*7*7
42,875	1000	5*5*5*7*7*7
60,025	1E00	5*5*7*7*7*7
78,125	1TS5	5*5*5*5*5*5*5
84,035	1YL0	5*7*7*7*7*7
109,375	2JA0	5*5*5*5*5*5*7
117,649	2R1E	7*7*7*7*7*7
153,125	3K00	5*5*5*5*5*7*7
214,375	5000	5*5*5*5*7*7*7
300,125	7000	5*5*5*7*7*7*7
390,625	93VQ	5*5*5*5*5*5*5*5
420,175	9T00	5*5*7*7*7*7*7
546,875	CRF0	5*5*5*5*5*5*5*7
588,245	DQ70	5*7*7*7*7*7*7
765,625	HV00	5*5*5*5*5*5*7*7
823,543	J79T	7*7*7*7*7*7*7
1,071,875	Q000	5*5*5*5*5*7*7*7
1,500,625	10000	5*5*5*5*7*7*7*7

As for primes with short recurring periods, there are 37 primes with pentatrigesimal periods of 20 or shorter, vis-à-vis 32 in decimal. One of these is as long as twenty decimal digits, another eighteen, and another sixteen, so I will not replicate all numbers with pentatrigesimal periods of 20 or shorter here. There are twenty primes with pentatrigesimal periods of 12 or shorter, vis-à-vis sixteen in decimal:

Pentatrigesimal	Pentatrigesimal repetend	Period	Decimal
2	0.Ḣ	1	2
H	0.2̇		17
3	0.ḂṄ	2	3
D	0.2̇P8̇	3	13
2S	0.0̇CṀ		97
HI	0.0̇1ZẎ	4	613
W	0.1̇4I29̇	5	31
15NR	0.0̇00V4̇		49,831
BC	0.0̇32ZWẊ	6	397
18	0.0̇TH38YḊ	7	43
UBENM	0.0̇00016Ṡ		44,007,727
HHHI	0.0̇001ZZZẎ	8	750,313
J	0.1̇UGK97CWḂ	9	19
1UGM3P	0.0̇0000IZZĠ		96,753,079
B	0.3̇6CQFWTM9J̇	10	11
339G	0.0̇00BAZZZNṖ		132,631
N	0.1̇I94JSDPC63̇	11	23
1AM8	0.0̇00RUI4JFCḊ		55,903
171RAE4	0.0̇00000U4LMṘ		2,208,546,869
7X	0.0̇4ESEMZVK7KĊ	12	277
4EN	0.0̇07X7WZZS2S3̇		5,413

Pentatrigesimal fractions for the first sixty natural numbers (periods listed in decimal notation) are:

	Pentatrigesimal reciprocal	Period	Decimal
1	1	0	1
2	0.Ḣ	1	2
3	0.ḂṄ	2	3
4	0.8̇Ṙ	2	4
5	0.7	0	5
6	0.5̇U̇	2	6
7	0.5	0	7
8	0.4̇Ḋ	2	8
9	0.3̇Ẇ	2	9
A	0.3Ḣ	1	10
B	0.3̇6CQFWTM9J̇	10	11
C	0.2̇Ẋ	2	12
D	0.2̇P8̇	3	13
E	0.2Ḣ	1	14
F	0.2ḂṄ	2	15
G	0.2̇6JṖ	4	16
H	0.2̇	1	17
I	0.1̇Ẏ	2	18
J	0.1̇UGK97CWḂ	9	19
K	0.1Ṙ8̇	2	20
L	0.1ṄḂ	2	21
M	0.1̇KNV7YEB4Ṡ	10	22
N	0.1̇I94JSDPC63̇	11	23
P	0.1̇Ġ	2	24
Q	0.1E	0	25
R	0.1̇C4̇	3	26
S	0.1̇ACYPṀ	6	27
T	0.18̇Ṙ	2	28
U	0.1̇78FP4TYSRJAV6̇	14	29
V	0.15̇U̇	2	30
W	0.1̇4I29̇	5	31
X	0.1̇39UIKSĊ	8	32
Y	0.1̇248GYXVRİ	10	33
Z	0.1̇	1	34
10	0.1	0	35
11	0.0̇Ż	2	36
12	0.0̇Y3SF4QIX5NMPKTD8HZ1W7JV9G2UBCAE6LRḢ	36	37
13	0.0̇X8A4L6FṄ	9	38
14	0.0̇WECJṘ	6	39
15	0.0V̇L̇	2	40
16	0.0̇UVQLBYA8ISB3EHXFCT5Z549DN1PRG7NWKH2JM6U̇	40	41
17	0.0U̇5̇	2	42
18	0.0̇TH38YḊ	7	43
19	0.0̇SUF3Z75JẆ	10	44
1A	0.0Ṡ7̇	2	45
1B	0.0̇RM29W6UNKJ̇	11	46
1C	0.0̇R286PK3QB5YHVILKUSJCN2Z8XRTAEW9NU1H4GDE57FMBẊ	46	47
1D	0.0̇QI8̇	4	48
1E	0.0Q	0	49
1F	0.0PḢ	1	50
1G	0.0̇Ṗ	2	51
1H	0.0̇NJI62̇	6	52
1I	0.0̇N3YNS2MFUQ3AJTDVD798KGHU1ZBW1B7XCJ59WPF6L4LSQREIH5Ẏ	52	53
1J	0.0̇MNZCḂ	6	54
1K	0.0Ṁ9J36CQFWṪ	10	55
1L	0.0L̇V̇	2	56
1M	0.0̇LH6RE4AFC9TV32FYṠ	18	57
1N	0.0̇L47UJWZDVS5Ḟ	14	58
1P	0.0̇KRPB9H745BV8WFEU2D1SA2YTGLCĠ	29	59
1Q	0.0K̇Ė	2	60

As you can see, the periods are different from those in decimal, apart from powers of five, the full period prime 47 (and 61 if I extended the list), the number 52, and the odd semiprimes 39 and 57. Noteworthy is the large number of denominators with pentatrigesimal period 2: nineteen numbers out of sixty, vis-à-vis only five in decimal.